As twistor59 already correctly wrote, a star and every other physical system may be found in a pure state in principle. When a physical system is in a pure state, it just means that we're assuming the maximum possible knowledge about the system – e.g. we know all the eigenvalues of a complete set of commuting observables.
If a physical system is in a mixed (non-pure) state, it just means that our knowledge about the system is smaller than the maximum one and we're creating a probabilistic mixture of pure states – which are mixed in a way that is fully analogous to probabilistic distributions in classical statistical physics.
In quantum mechanics, predictions are always probabilistic but for any pure state, the role for the probabilities is "minimized". Pure states are the closest objects somewhat analogous to "points in the phase space" in classical physics.
In the black hole information puzzle research and many other considerations, we're considering the initial state of the matter to be in a pure state and we can do so without a loss of generality because the evolution of any mixed state is completely determined by the evolution of pure states that belong to a basis of the Hilbert space (space of possible pure states). The previous sentence, easily proved from the evolution equations for density matrices and pure states, is just the quantum counterpart of the classical statement that it's enough to investigate the behavior of well-defined points in the phase space (pure classical states) and the behavior of all probabilistic mixtures is then determined by purely statistical (Hamiltonian-independent) formulae.
A star of a certain temperature may be described by a mixed state but in principle, we may also assume that the actual state of the star is a pure state that just "resembles" the mixed state – a typical pure state in the ensemble, for example – and we may study the behavior of these pure states separately. Indeed, that's necessary if we want the potential information loss to be seen at all. The information loss was the apparent conclusion by Hawking in the 1970s that a pure state will ultimately evolve into a mixed one. There has never been a doubt that a mixed state evolves into a mixed state but the problem was that even a pure state seemed not to be willing to stay pure. This conclusion was later shown to be an artifact of the perturbative approximations in quantum gravity. The full theory of quantum gravity evolves pure states into pure states of the final Hawking radiation.